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Unformatted text preview: Modern Analysis, Homework 3 1. Recall that a function f on an interval [ a,b ] is of bounded variation if sup n N x = a<x 1 < <x n = b n X i =0  f ( x i ) f ( x i +1 )  < + . In such a case, we write f BV[ a,b ]. Given f BV[ a,b ], for x [ a,b ], define T f ( x ) = sup n N x = a<x 1 < <x n = x n X i =0  f ( x i ) f ( x i +1 )  (the total variation of f on [ a,x ]). (a) Prove that T f ( x ) + f ( x ) and T f ( x ) f ( x ) are increasing functions on [ a,b ]. (b) Prove that f BV[ a,b ] if and only if f can be written as a difference of increasing functions. 2. Let { f n } n N be a sequence of continuous functions defined on a compact metric space ( K,d ). Suppose that for every convergent sequence of points { x n } n N such that x n x , we have lim n f n ( x n ) = f ( x ) where f is some continuous function on K . Prove that f n f uniformly on K . 3. Let g be a continuous function on [0 , 1] with g (1) = 0. Prove that the sequence { g ( x ) x n } converges uniformly on [0 , 1]. 4. (a) Suppose that f is a 2 periodic function on R , Riemannintegrable on [ , ]....
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This note was uploaded on 10/18/2011 for the course MATH S4062Q taught by Professor Staff during the Summer '11 term at Columbia.
 Summer '11
 Staff

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